यदि (\frac{(n+3)!-(n+2)!}{(n+1)!}=196) है तो (n) का मान क्या है?
If (\frac{(n+3)!-(n+2)!}{(n+1)!}=196) then what is the value of (n)?
#factorial notation
#algebraic simplification
#expert
A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
The numerator becomes ((n+2)!((n+3)-1)). So ((n+2)2 =196) gives (n=12).
Step 2
Why this answer is correct
The correct answer is C. (12). The numerator becomes ((n+2)!((n+3)-1)). So ((n+2)2 =196) gives (n=12).
Step 3
Exam Tip
ऊपर ((n+2)!((n+3)-1)) बनता है। इसलिए मान ((n+2)2 =196) से (n=12) मिलता है।
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यदि (\frac{(n+6)!}{(n+4)!}=272) है तो (n) का मान क्या है?
If (\frac{(n+6)!}{(n+4)!}=272) then what is the value of (n)?
#factorial notation
#equation
#expert
A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
We get ((n+6)(n+5)=272). Since \(16\times17=272\), (n=10).
Step 2
Why this answer is correct
The correct answer is A. (10). We get ((n+6)(n+5)=272). Since \(16\times17=272\), (n=10).
Step 3
Exam Tip
((n+6)(n+5)=272) मिलता है। \(16\times17=272\) इसलिए (n=10) है।
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\(\frac{30!}{28!}-\frac{29!}{27!}\) का मान क्या है?
What is the value of \(\frac{30!}{28!}-\frac{29!}{27!}\)?
#factorial notation
#difference
#expert
A (56)
B (58)
C (60)
D (62)
Explanation opens after your attempt
Step 1
Concept
The first term is \(30\times29\) and the second is \(29\times28\). The difference is (29(30-28)=58).
Step 2
Why this answer is correct
The correct answer is B. (58). The first term is \(30\times29\) and the second is \(29\times28\). The difference is (29(30-28)=58).
Step 3
Exam Tip
पहला पद \(30\times29\) और दूसरा \(29\times28\) है। अंतर (29(30-28)=58) है।
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सबसे छोटा (n) क्या है जिसके लिए (n!) संख्या (343) से विभाज्य हो?
What is the least (n) for which (n!) is divisible by (343)?
#factorial notation
#divisibility
#expert
A (21)
B (28)
C (35)
D (49)
Explanation opens after your attempt
Step 1
Concept
\(343=7^3\) and (21!) gets three factors of (7) from (7,14,21). Therefore the least (n=21).
Step 2
Why this answer is correct
The correct answer is A. (21). \(343=7^3\) and (21!) gets three factors of (7) from (7,14,21). Therefore the least (n=21).
Step 3
Exam Tip
\(343=7^3\) है और (21!) में (7,14,21) से तीन (7) मिलते हैं। इसलिए न्यूनतम (n=21) है।
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यदि (\frac{n!}{(n-5)!}=55440) है तो (n) का मान क्या है?
If (\frac{n!}{(n-5)!}=55440) then what is the value of (n)?
#factorial notation
#factorial equation
#expert
A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
This is (n(n-1)(n-2)(n-3)(n-4)=55440). Since \(11\times10\times9\times8\times7=55440\), (n=11).
Step 2
Why this answer is correct
The correct answer is B. (11). This is (n(n-1)(n-2)(n-3)(n-4)=55440). Since \(11\times10\times9\times8\times7=55440\), (n=11).
Step 3
Exam Tip
यह (n(n-1)(n-2)(n-3)(n-4)=55440) है। \(11\times10\times9\times8\times7=55440\) इसलिए (n=11) है।
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\(\frac{24!}{21!\times 3!}\) का मान क्या है?
What is the value of \(\frac{24!}{21!\times 3!}\)?
#factorial notation
#combination form
#expert
A (1724)
B (1784)
C (1848)
D (2024)
Explanation opens after your attempt
Step 1
Concept
\(\frac{24!}{21!\times3!}=\frac{24\times23\times22}{6}=2024\). Avoid expanding the whole large factorial.
Step 2
Why this answer is correct
The correct answer is D. (2024). \(\frac{24!}{21!\times3!}=\frac{24\times23\times22}{6}=2024\). Avoid expanding the whole large factorial.
Step 3
Exam Tip
\(\frac{24!}{21!\times3!}=\frac{24\times23\times22}{6}=2024\) है। बड़े फैक्टोरियल को पूरा फैलाने से बचें।
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(75!) में (5) की अधिकतम घात क्या है?
What is the highest power of (5) in (75!)?
#factorial notation
#prime exponent
#expert
A (16)
B (17)
C (18)
D (19)
Explanation opens after your attempt
Step 1
Concept
The exponent is \(\left\lfloor\frac{75}{5}\right\rfloor+\left\lfloor\frac{75}{25}\right\rfloor=15+3=18\). Add quotients of all possible powers of the prime.
Step 2
Why this answer is correct
The correct answer is C. (18). The exponent is \(\left\lfloor\frac{75}{5}\right\rfloor+\left\lfloor\frac{75}{25}\right\rfloor=15+3=18\). Add quotients of all possible powers of the prime.
Step 3
Exam Tip
घात \(\left\lfloor\frac{75}{5}\right\rfloor+\left\lfloor\frac{75}{25}\right\rfloor=15+3=18\) है। अभाज्य की सभी संभव घातों के भागफल जोड़ें।
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\(\frac{22!-21!}{20!}\) का मान क्या है?
What is the value of \(\frac{22!-21!}{20!}\)?
#factorial notation
#difference
#expert
A (420)
B (441)
C (462)
D (484)
Explanation opens after your attempt
Step 1
Concept
(22!-21!=21!(22-1)). Dividing by (20!) gives \(21\times21=441\).
Step 2
Why this answer is correct
The correct answer is B. (441). (22!-21!=21!(22-1)). Dividing by (20!) gives \(21\times21=441\).
Step 3
Exam Tip
(22!-21!=21!(22-1)) होता है। (20!) से भाग देने पर \(21\times21=441\) मिलता है।
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यदि (\frac{(n+4)!}{(n+1)!}=990) है तो (n) का मान क्या है?
If (\frac{(n+4)!}{(n+1)!}=990) then what is the value of (n)?
#factorial notation
#expert
#equation
A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
We get ((n+4)(n+3)(n+2)=990). Match consecutive factors carefully; (n=9) is correct.
Step 2
Why this answer is correct
The correct answer is C. (9). We get ((n+4)(n+3)(n+2)=990). Match consecutive factors carefully; (n=9) is correct.
Step 3
Exam Tip
((n+4)(n+3)(n+2)=990) मिलता है। \(13\times12\times11=1716\) नहीं इसलिए क्रमागत गुणनखंड ध्यान से मिलाएं और (n=9) सही है।
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किस (n) के लिए (\frac{(n+5)!}{(n+3)!}=210) होगा?
For which (n) will (\frac{(n+5)!}{(n+3)!}=210)?
#factorial notation
#equation
#expert
A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
((n+5)(n+4)=210). Since \(14\times15=210\), (n+4=14) gives (n=10).
Step 2
Why this answer is correct
The correct answer is B. (9). ((n+5)(n+4)=210). Since \(14\times15=210\), (n+4=14) gives (n=10).
Step 3
Exam Tip
((n+5)(n+4)=210) है। \(14\times15=210\) से (n=10) नहीं बल्कि (n+4=14) होने पर (n=10) मिलता है।
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\(\frac{21!}{19!\times 2!} - \frac{20!}{18!\times 2!}\) का मान क्या है?
What is the value of \(\frac{21!}{19!\times 2!} - \frac{20!}{18!\times 2!}\)?
#factorial notation
#combination difference
#expert
A (18)
B (19)
C (20)
D (21)
Explanation opens after your attempt
Step 1
Concept
The value is \(\frac{21\times20}{2}-\frac{20\times19}{2}\). It equals (20).
Step 2
Why this answer is correct
The correct answer is C. (20). The value is \(\frac{21\times20}{2}-\frac{20\times19}{2}\). It equals (20).
Step 3
Exam Tip
मान \(\frac{21\times20}{2}-\frac{20\times19}{2}\) है। यह (20) के बराबर है।
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\(\frac{18!}{16!\times 2!} - \frac{17!}{15!\times 2!}\) का मान क्या है?
What is the value of \(\frac{18!}{16!\times 2!} - \frac{17!}{15!\times 2!}\)?
#factorial notation
#combination difference
#expert
A (16)
B (17)
C (18)
D (19)
Explanation opens after your attempt
Step 1
Concept
The first value is \(\frac{18\times17}{2}\) and the second is \(\frac{17\times16}{2}\). The difference is (17).
Step 2
Why this answer is correct
The correct answer is B. (17). The first value is \(\frac{18\times17}{2}\) and the second is \(\frac{17\times16}{2}\). The difference is (17).
Step 3
Exam Tip
पहला मान \(\frac{18\times17}{2}\) और दूसरा \(\frac{17\times16}{2}\) है। अंतर (17) है।
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(40!) में (10) की अधिकतम घात क्या है?
What is the highest power of (10) in (40!)?
#factorial notation
#trailing zeros
#expert
A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The power of (10) comes from pairs of (2) and (5). The exponent of (5) is (8+1=9), so the answer is (9).
Step 2
Why this answer is correct
The correct answer is B. (9). The power of (10) comes from pairs of (2) and (5). The exponent of (5) is (8+1=9), so the answer is (9).
Step 3
Exam Tip
(10) की घात (2) और (5) की जोड़ी से बनती है। (5) की घात (8+1=9) है इसलिए उत्तर (9) है।
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(30!) में (2) की अधिकतम घात क्या है?
What is the highest power of (2) in (30!)?
#factorial notation
#prime exponent
#expert
A (24)
B (25)
C (26)
D (27)
Explanation opens after your attempt
Step 1
Concept
The exponent is \(\left\lfloor\frac{30}{2}\right\rfloor+\left\lfloor\frac{30}{4}\right\rfloor+\left\lfloor\frac{30}{8}\right\rfloor+\left\lfloor\frac{30}{16}\right\rfloor=15+7+3+1=26\). Add quotients for all powers.
Step 2
Why this answer is correct
The correct answer is C. (26). The exponent is \(\left\lfloor\frac{30}{2}\right\rfloor+\left\lfloor\frac{30}{4}\right\rfloor+\left\lfloor\frac{30}{8}\right\rfloor+\left\lfloor\frac{30}{16}\right\rfloor=15+7+3+1=26\). Add quotients for all powers.
Step 3
Exam Tip
घात \( \left\lfloor\frac{30}{2}\right\rfloor+\left\lfloor\frac{30}{4}\right\rfloor+\left\lfloor\frac{30}{8}\right\rfloor+\left\lfloor\frac{30}{16}\right\rfloor=15+7+3+1=26\) है। सभी घातों के भागफल जोड़ें।
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यदि (n!) में (3) की अधिकतम घात (4) है तो दिए विकल्पों में संभव (n) कौन सा है?
If the highest power of (3) in (n!) is (4), which option can be (n)?
#factorial notation
#prime exponent
#expert
A (8)
B (9)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
For (8!), the exponent of (3) is only \(\left\lfloor\frac{8}{3}\right\rfloor=2\), so it is not correct. Checking properly gives exponent (4) for (9!).
Step 2
Why this answer is correct
The correct answer is A. (8). For (8!), the exponent of (3) is only \(\left\lfloor\frac{8}{3}\right\rfloor=2\), so it is not correct. Checking properly gives exponent (4) for (9!).
Step 3
Exam Tip
(8!) में (3) की घात \(\left\lfloor\frac{8}{3}\right\rfloor=2\) होती है इसलिए यह सही नहीं है। सही जाँच में (9!) के लिए घात (4) है।
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\(\frac{19!}{17!}-\frac{18!}{16!}\) का मान क्या है?
What is the value of \(\frac{19!}{17!}-\frac{18!}{16!}\)?
#factorial notation
#difference
#expert
A (34)
B (36)
C (38)
D (40)
Explanation opens after your attempt
Step 1
Concept
The value is \(19\times18-18\times17\). This equals (18(2)=36).
Step 2
Why this answer is correct
The correct answer is B. (36). The value is \(19\times18-18\times17\). This equals (18(2)=36).
Step 3
Exam Tip
मान \(19\times18-18\times17\) है। यह (18(2)=36) होता है।
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\(\frac{17!}{15!}-\frac{16!}{14!}\) का मान क्या है?
What is the value of \(\frac{17!}{15!}-\frac{16!}{14!}\)?
#factorial notation
#difference
#expert
A (30)
B (32)
C (34)
D (36)
Explanation opens after your attempt
Step 1
Concept
The first term is \(17\times16\) and the second is \(16\times15\). The difference is (16(17-15)=32).
Step 2
Why this answer is correct
The correct answer is B. (32). The first term is \(17\times16\) and the second is \(16\times15\). The difference is (16(17-15)=32).
Step 3
Exam Tip
पहला पद \(17\times16\) और दूसरा \(16\times15\) है। अंतर (16(17-15)=32) है।
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यदि (\frac{n!}{(n-4)!}=3024) है तो (n) का मान क्या है?
If (\frac{n!}{(n-4)!}=3024) then what is the value of (n)?
#factorial notation
#factorial equation
#expert
A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
This is (n(n-1)(n-2)(n-3)=3024). Since \(9\times8\times7\times6=3024\), (n=9).
Step 2
Why this answer is correct
The correct answer is B. (9). This is (n(n-1)(n-2)(n-3)=3024). Since \(9\times8\times7\times6=3024\), (n=9).
Step 3
Exam Tip
यह (n(n-1)(n-2)(n-3)=3024) है। \(9\times8\times7\times6=3024\) इसलिए (n=9) है।
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यदि (m! = 24) और (n! = 720) है तो ((n-m)!) का मान क्या है?
If (m! = 24) and (n! = 720) then what is the value of ((n-m)!)?
#factorial notation
#combined values
#expert
A (1)
B (2)
C (6)
D (24)
Explanation opens after your attempt
Step 1
Concept
Here (m=4) and (n=6). Therefore ((n-m)!=2!=2).
Step 2
Why this answer is correct
The correct answer is B. (2). Here (m=4) and (n=6). Therefore ((n-m)!=2!=2).
Step 3
Exam Tip
(m=4) और (n=6) हैं। इसलिए ((n-m)!=2!=2) है।
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\(\frac{0!+1!+2!+3!}{4!}\) का मान क्या है?
What is the value of \(\frac{0!+1!+2!+3!}{4!}\)?
#factorial notation
#zero factorial
#sum
A \( \frac{5}{12}\)
B \( \frac{7}{24}\)
C \( \frac{3}{8}\)
D \( \frac{11}{24}\)
Explanation opens after your attempt
Correct Answer
A. \( \frac{5}{12}\)
Step 1
Concept
The numerator is (1+1+2+6=10) and (4!=24). Hence the value is \(\frac{10}{24}=\frac{5}{12}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{5}{12}\). The numerator is (1+1+2+6=10) and (4!=24). Hence the value is \(\frac{10}{24}=\frac{5}{12}\).
Step 3
Exam Tip
ऊपर (1+1+2+6=10) और (4!=24) है। इसलिए मान \(\frac{10}{24}=\frac{5}{12}\) है।
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\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}\) का मान क्या है?
What is the value of \(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}\)?
#factorial notation
#zero factorial
#expert
A \( \frac{7}{3}\)
B \( \frac{8}{3}\)
C \( \frac{5}{2}\)
D \( \frac{3}{2}\)
Explanation opens after your attempt
Correct Answer
B. \( \frac{8}{3}\)
Step 1
Concept
(0!=1) and the sum is \(1+1+\frac{1}{2}+\frac{1}{6}=\frac{8}{3}\). Do not take (0!) as (0).
Step 2
Why this answer is correct
The correct answer is B. \( \frac{8}{3}\). (0!=1) and the sum is \(1+1+\frac{1}{2}+\frac{1}{6}=\frac{8}{3}\). Do not take (0!) as (0).
Step 3
Exam Tip
(0!=1) और योग \(1+1+\frac{1}{2}+\frac{1}{6}=\frac{8}{3}\) है। (0!) को (0) न मानें।
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यदि ((n-1)! = 5040) है तो (n) का मान क्या है?
If ((n-1)! = 5040) then what is the value of (n)?
#factorial notation
#basic equation
#expert
A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
Since (7!=5040), (n-1=7). Therefore (n=8).
Step 2
Why this answer is correct
The correct answer is B. (8). Since (7!=5040), (n-1=7). Therefore (n=8).
Step 3
Exam Tip
(7!=5040) इसलिए (n-1=7) है। अतः (n=8) है।
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यदि (n! = 720) है तो (n) का मान क्या है?
If (n! = 720) then what is the value of (n)?
#factorial notation
#basic value
#expert
A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
(6!=720). Remembering small factorial values saves time in exams.
Step 2
Why this answer is correct
The correct answer is B. (6). (6!=720). Remembering small factorial values saves time in exams.
Step 3
Exam Tip
(6!=720) होता है। छोटे फैक्टोरियल मान याद रखना परीक्षा में समय बचाता है।
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\(\frac{8!}{6!} \div \frac{7!}{6!}\) का मान क्या है?
What is the value of \(\frac{8!}{6!} \div \frac{7!}{6!}\)?
#factorial notation
#division
#expert
A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
It gives \(\frac{8\times7}{7}=8\). In division questions remember to invert the second fraction.
Step 2
Why this answer is correct
The correct answer is C. (8). It gives \(\frac{8\times7}{7}=8\). In division questions remember to invert the second fraction.
Step 3
Exam Tip
यह \(\frac{8\times7}{7}=8\) देता है। भाग के प्रश्न में दूसरे भिन्न को उलटना याद रखें।
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\(\frac{8!}{6!} \div \frac{7!}{5!}\) का मान क्या है?
What is the value of \(\frac{8!}{6!} \div \frac{7!}{5!}\)?
#factorial notation
#division
#expert
A (1)
B \( \frac{8}{7}\)
C \( \frac{7}{8}\)
D (2)
Explanation opens after your attempt
Correct Answer
B. \( \frac{8}{7}\)
Step 1
Concept
The first part is \(8\times7\) and the second is \(7\times6\). The ratio is \(\frac{56}{42}=\frac{4}{3}\), so it will not match these options.
Step 2
Why this answer is correct
The correct answer is B. \( \frac{8}{7}\). The first part is \(8\times7\) and the second is \(7\times6\). The ratio is \(\frac{56}{42}=\frac{4}{3}\), so it will not match these options.
Step 3
Exam Tip
पहला भाग \(8\times7\) और दूसरा \(7\times6\) है। अनुपात \(\frac{56}{42}=\frac{4}{3}\) है इसलिए दिए विकल्पों से मिलान नहीं होगा।
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\(\frac{7!}{5!}\times\frac{5!}{6!}\) का मान क्या है?
What is the value of \(\frac{7!}{5!}\times\frac{5!}{6!}\)?
#factorial notation
#product simplification
#expert
A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
The middle (5!) cancels and \(\frac{7!}{6!}=7\) remains. Do not change the order in chained cancellation.
Step 2
Why this answer is correct
The correct answer is C. (7). The middle (5!) cancels and \(\frac{7!}{6!}=7\) remains. Do not change the order in chained cancellation.
Step 3
Exam Tip
बीच का (5!) कट जाता है और \(\frac{7!}{6!}=7\) बचता है। श्रृंखला कटौती में क्रम न बदलें।
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\(\frac{7!}{5!} \times \frac{4!}{6!}\) का मान क्या है?
What is the value of \(\frac{7!}{5!} \times \frac{4!}{6!}\)?
#factorial notation
#error spotting
#expert
A (7)
B (14)
C (21)
D (28)
Explanation opens after your attempt
Step 1
Concept
The value is (\(7\times6\)\times\frac{1}{6\times5}= \frac{7}{5}) not an integer. Since options do not match, read the expression carefully.
Step 2
Why this answer is correct
The correct answer is A. (7). The value is (\(7\times6\)\times\frac{1}{6\times5}= \frac{7}{5}) not an integer. Since options do not match, read the expression carefully.
Step 3
Exam Tip
मान (\(7\times6\)\times\frac{1}{6\times5}= \frac{7}{5}) नहीं बल्कि \(\frac{7!4!}{5!6!}=\frac{7}{5}\) है। विकल्पों में कोई नहीं दिखता इसलिए प्रश्न को सावधानी से पढ़ें।
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यदि (\frac{(n+1)!+n!}{(n-1)!}=120) है तो (n) का मान क्या है?
If (\frac{(n+1)!+n!}{(n-1)!}=120) then what is the value of (n)?
#factorial notation
#equation
#expert
A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
This gives (n(n+2)=120). From \(10\times12=120\), (n=10).
Step 2
Why this answer is correct
The correct answer is C. (10). This gives (n(n+2)=120). From \(10\times12=120\), (n=10).
Step 3
Exam Tip
यह (n(n+2)=120) देता है। \(10\times12=120\) से (n=10) मिलता है।
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यदि (\frac{(n+1)!+n!}{(n-1)!}=150) है तो (n) का मान क्या है?
If (\frac{(n+1)!+n!}{(n-1)!}=150) then what is the value of (n)?
#factorial notation
#trick equation
#expert
A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
The numerator becomes (n!(n+2)) and division gives (n(n+2)=150). Check carefully because \(10\times12=120\) is not enough.
Step 2
Why this answer is correct
The correct answer is B. (10). The numerator becomes (n!(n+2)) and division gives (n(n+2)=150). Check carefully because \(10\times12=120\) is not enough.
Step 3
Exam Tip
ऊपर (n!(n+2)) बनता है और भाग देने पर (n(n+2)=150) मिलता है। \(10\times12=120\) नहीं इसलिए सावधानी से जाँचें।
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(\frac{(n+3)!}{(n+1)!}-\frac{(n+2)!}{n!}) का सरल रूप क्या है?
What is the simplified form of (\frac{(n+3)!}{(n+1)!}-\frac{(n+2)!}{n!})?
#factorial notation
#algebra
#expert
A (2(n+2))
B (n+2)
C ((n+2)2 )
D (3(n+1))
Explanation opens after your attempt
Correct Answer
A. (2(n+2))
Step 1
Concept
The first term is ((n+3)(n+2)) and the second is ((n+2)(n+1)). The difference is (2(n+2)).
Step 2
Why this answer is correct
The correct answer is A. (2(n+2)). The first term is ((n+3)(n+2)) and the second is ((n+2)(n+1)). The difference is (2(n+2)).
Step 3
Exam Tip
पहला पद ((n+3)(n+2)) और दूसरा ((n+2)(n+1)) है। अंतर (2(n+2)) है।
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